Download SECTION 1-2 Polynomials: Basic Operations

12
1 Basic Algebraic Operations
57. If c 0.151515 . . . , then 100c 15.1515 . . . and
59. To see how the distributive property is behind the mechanics of long multiplication, compute each of the following
and compare:
100c c 15.1515 . . . 0.151515 . . .
99c 15
Long
Multiplication
23
12
5
c 15
99 33
Proceeding similarly, convert the repeating decimal
0.090909 . . . into a fraction. (All repeating decimals are rational numbers, and all rational numbers have repeating
decimal representations.)
60. For a and b real numbers, justify each step using a property
in this section.
58. Repeat Problem 57 for 0.181818. . . .
SECTION
1-2
Use of the
Distributive Property
23 12
23(2 10)
23 2 23 10 Statement
1. (a b) (a) (a) (a b)
2.
[(a) a] b
3.
0b
4.
b
Reason
1.
2.
3.
4.
Polynomials: Basic Operations
•
•
•
•
•
•
•
Natural Number Exponents
Polynomials
Combining Like Terms
Addition and Subtraction
Multiplication
Combined Operations
Application
In this section we review the basic operations on polynomials, a mathematical form
encountered frequently throughout mathematics. We start the discussion with a brief
review of natural number exponents. Integer and rational exponents and their properties will be discussed in detail in subsequent sections.
• Natural Number
The definition of a natural number exponent is given below:
Exponents
DEFINITION 1
Natural Number Exponent
For n a natural number and a any real number:
an a a · · · a
agfgbggfc
n factors of a
24 2 2 2 2
4 factors of 2
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Polynomials: Basic Operations
13
Also, the first property of exponents is stated as follows:
Theorem 1
First Property of Exponents
For any natural numbers m and n, and any real number a:
aman am n
• Polynomials
(3x5)(2x7)
(3 2)x57
*
6x12
Algebraic expressions are formed by using constants and variables and the algebraic
operations of addition, subtraction, multiplication, division, raising to powers, and taking roots. Some examples are
3
x3 5
5x4 2x2 7
xy7
(2x y)2
x5
x 2x 5
1
2
1
1
1
x
An algebraic expression involving only the operations of addition, subtraction,
multiplication, and raising to natural number powers on variables and constants is
called a polynomial. Some examples are
2x 3
4x2 3x 7
x 2y
5x3 2x2 7x 9
5
x2 3xy 4y2
0
x3 3x2y xy2 2y7
In a polynomial, a variable cannot appear in a denominator, as an exponent, or within
a radical. Accordingly, a polynomial in one variable x is constructed by adding or
subtracting constants and terms of the form axn, where a is a real number and n is a
natural number. A polynomial in two variables x and y is constructed by adding and
subtracting constants and terms of the form axmyn, where a is a real number and m
and n are natural numbers. Polynomials in three or more variables are defined in a
similar manner.
Polynomial forms can be classified according to their degree. If a term in a polynomial has only one variable as a factor, then the degree of that term is the power
of the variable. If two or more variables are present in a term as factors, then the
degree of the term is the sum of the powers of the variables. The degree of a polynomial is the degree of the nonzero term with the highest degree in the polynomial.
Any nonzero constant is defined to be a polynomial of degree 0. The number 0 is
also a polynomial but is not assigned a degree.
*Throughout the book, dashed boxes—called think boxes—are used to represent steps that are usually
performed mentally.
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1 Basic Algebraic Operations
EXAMPLE 1
Polynomials and Nonpolynomials
(A) Polynomials in one variable:
x2 3x 2
6x3 2x 13
(B) Polynomials in several variables:
3x2 2xy y2
4x3y2 3xy2z5
(C) Nonpolynomials:
2x x2 3x 2
x3
3
5
x
x2 3x 1
(D) The degree of the first term in 6x3 2x 13 is 3, the degree of the second
term is 1, the degree of the third term is 0, and the degree of the whole polynomial is 3.
(E) The degree of the first term in 4x3y2 3xy2 is 5, the degree of the second
term is 3, and the degree of the whole polynomial is 5.
Matched Problem 1
(A) Which of the following are polynomials?
3x2 2x 1
x 3
x2 2xy y2
x1
x2 2
(B) Given the polynomial 3x5 6x3 5, what is the degree of the first term? The
second term? The whole polynomial?
(C) Given the polynomial 6x4y2 3xy3, what is the degree of the first term? The
second term? The whole polynomial?
In addition to classifying polynomials by degree, we also call a single-term polynomial a monomial, a two-term polynomial a binomial, and a three-term polynomial
a trinomial.
5 2 3
2x y
Monomial
x 4.7
Binomial
x4 2x2 9
Trinomial
3
• Combining
Like Terms
We start with a word about coefficients. A constant in a term of a polynomial,
including the sign that precedes it, is called the numerical coefficient, or simply,
the coefficient, of the term. If a constant doesn’t appear, or only a sign appears,
1-2
Polynomials: Basic Operations
15
the coefficient is understood to be 1. If only a sign appears, the coefficient is understood to be 1. Thus, given the polynomial
2x4 4x3 x2 x 5
2x4 (4)x3 1x2 (1)x 5
the coefficient of the first term is 2, the coefficient of the second term is 4, the coefficient of the third term is 1, the coefficient of the fourth term is 1, and the coefficient of the last term is 5.
At this point, it is useful to state two additional distributive properties of real
numbers that follow from the distributive properties stated in Section 1-1.
Additional Distributive Properties
1. a(b c) (b c)a ab ac
2. a(b c · · · f ) ab ac · · · af
Two terms in a polynomial are called like terms if they have exactly the same
variable factors to the same powers. The numerical coefficients may or may not be the
same. Since constant terms involve no variables, all constant terms are like terms. If
a polynomial contains two or more like terms, these terms can be combined into a single term by making use of distributive properties. Consider the following example:
5x3y 2xy x3y 2x3y
5x3y x3y 2x3y 2xy
(5x3y x3y 2x3y) 2xy
(5 1 2)x3y 2xy
2x3y 2xy
It should be clear that free use has been made of the real number properties discussed earlier. The steps done in the dashed box are usually done mentally, and the
process is quickly mechanized as follows:
Like terms in a polynomial are combined by adding their numerical
coefficients.
EXAMPLE 2
Simplifying Polynomials
Remove parentheses and combine like terms:
(A) 2(3x2 2x 5) (x2 3x 7)
2(3x2 2x 5) 1(x2 3x 7)
Think
6x2 4x 10 x2 3x 7
7x2 x 3
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1 Basic Algebraic Operations
(B) (x3 2x 6) (2x3 x2 2x 3)
1(x3 2x 6) (1)(2x3 x2 2x 3)
Think
Be careful with
the sign here.
x3 2x 6 2x3 x2 2x 3
x3 x2 4x 3
(C) [3x2 (2x 1)] (x2 1) [3x2 2x 1] (x2 1)
3x2 2x 1 x2 1
Remove inner
parentheses first.
2x2 2x
Matched Problem 2
Remove parentheses and combine like terms:
(A) 3(u2 2v2) (u2 5v2)
(B) (m3 3m2 m 1) (2m3 m 3)
(C) (x3 2) [2x3 (3x 4)]
• Addition and
Subtraction
EXAMPLE 3
Addition and subtraction of polynomials can be thought of in terms of removing
parentheses and combining like terms, as illustrated in Example 2. Horizontal and vertical arrangements are illustrated in the next two examples. You should be able to
work either way, letting the situation dictate the choice.
Adding Polynomials
Add: x4 3x3 x2,
Solution
x3 2x2 3x,
and
3x2 4x 5
Add horizontally:
(x4 3x3 x2) (x3 2x2 3x) (3x2 4x 5)
x4 3x3 x2 x3 2x2 3x 3x2 4x 5
x4 4x3 2x2 x 5
Or vertically, by lining up like terms and adding their coefficients:
x4 3x3 x2
x4 x3 2x2 3x
x4 x3 3x2 4x 5
x4 4x3 2x2 x 5
1-2
Matched Problem 3
Matched Problem 4
CAUTION
x3 2x2 5x,
4x2 3x 5
(x2 8) (4x2 3x 5)
or
EXAMPLE 5
x2 8
x2 8 4x2 3x 5
4x2 3x 5
3x2 3x 13
3x2 3x 13
Subtract:
2x2 5x 4
from
← Change signs and add.
5x2 6
When you use a horizontal arrangement to subtract a polynomial with more
than one term, you must enclose the polynomial in parentheses. Thus, to
subtract 2x 5 from 4x 11, you must write
and not
4x 11 2x 5
Multiplication of algebraic expressions involves the extensive use of distributive properties for real numbers, as well as other real number properties.
Multiplying Polynomials
Multiply:
Solution
from
4x2 3x 8
4x 11 (2x 5)
• Multiplication
x2 7x 2
and
Subtracting Polynomials
Subtract:
Solution
17
Add horizontally and vertically:
3x4 2x3 4x2,
EXAMPLE 4
Polynomials: Basic Operations
(2x 3)(3x2 2x 3)
(2x 3)(3x2 2x 3)
2x(3x2 2x 3) 3(3x2 2x 3)
6x3 4x2 6x 9x2 6x 9
6x3 13x2 12x 9
Or, using a vertical arrangement,
3x2 2x 3
2x2 3
6x3 4x2 6x
6x3 9x2 6x 9
6x3 13x2 12x 9
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1 Basic Algebraic Operations
Matched Problem 5
(2x 3)(2x2 3x 2)
Multiply:
Thus, to multiply two polynomials, multiply each term of one by each term of
the other, and combine like terms.
Products of certain binomial factors occur so frequently that it is useful to develop
procedures that will enable us to write down their products by inspection. To find the
product (2x 1)(3x 2), we will use the popular FOIL method. We multiply each
term of one factor by each term of the other factor as follows:
F
First
product
↓
(2x 1)(3x 2) 6x2
O
I
L
Outer
Inner
Last
product product product
↓
4x
↓
3x
↓
2
The inner and outer products are like terms and hence combine into one term. Thus,
(2x 1)(3x 2) 6x2 x 2
To speed up the process, we combine the inner and outer product mentally.
Products of certain binomial factors occur so frequently that it is useful to remember formulas for their products. The following formulas are easily verified by multiplying the factors on the left using the FOIL method:
Special Products
1. (a b)(a b) a2 b2
2. (a b)2 a2 2ab b2
3. (a b)2 a2 2ab b2
EXPLORE-DISCUSS 1
(A) Explain the relationship between special product formula 1 and the areas of
the rectangles in the figures.
(a b)(a b)
a2 b2
a
ab
a
a
b
b
b
(B) Construct similar figures to provide geometric interpretations for special product formulas 2 and 3.
1-2
EXAMPLE 6
Polynomials: Basic Operations
19
Multiplying Binomials
Multiply:
(A) (2x 3y)(5x 2y)
10x2 4xy 15xy 6y2
(B) (3a 2b)(3a 2b)
(3a)2 (2b)2
(C) (5x 3)2
(5x)2 2(5x)(3) 32
10x2 11xy 6y2
9a2 4b2
25x2 30x 9
(D) (m 2n)2 m2 4mn 4n2
Matched Problem 6
Multiply:
(A) (4u 3v)(2u v)
(C) (m 4n)(m 4n)
CAUTION
(B) (2xy 3)(2xy 3)
(D) (2u 3v)2
Remember to include the sum of the inner and outer terms when using the
FOIL method to square a binomial. That is,
(x 3)2 x2 9
• Combined
Operations
(E) (6x y)2
(x 3)2 x2 6x 9
We now consider several examples that use all the operations just discussed. Before
considering these examples, it is useful to summarize order-of-operation conventions
pertaining to exponents, multiplication and division, and addition and subtraction.
Order of Operations
1. Simplify inside the innermost grouping first, then the next innermost, and so on.
2[3 (x 4)] 2[3 x 4]
2(7 x) 14 2x
2. Unless grouping symbols indicate otherwise, apply exponents before multiplication or division is performed.
2(x 2)2 2(x2 4x 4) 2x2 8x 8
3. Unless grouping symbols indicate otherwise, perform multiplication and
division before addition and subtraction. In either case, proceed from left to
right.
5 2(x 3) 5 2x 6 11 2x
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1 Basic Algebraic Operations
EXAMPLE 7
Combined Operations
Perform the indicated operations and simplify:
(A) 3x {5 3[x x(3 x)]} 3x {5 3[x 3x x2]}
3x {5 3[2x x2]}
3x {5 6x 3x2}
3x 5 6x 3x2
3x2 3x 5
(B) (x 2y)(2x 3y) (2x y)2 2x2 3xy 4xy 6y2 (4x2 4xy y2)
2x2 xy 6y2 4x2 4xy y2
2x2 5xy 7y2
(C) (2m 3n)3 (2m 3n)(2m 3n)2
(2m 3n)(4m2 12mn 9n2)
8m3 24m2n 18mn2 12m2n 36mn2 27n3
8m3 36m2n 54mn2 27n3
Matched Problem 7
Perform the indicated operations and simplify:
(A) 2t {7 2[t t(4 t)]}
(C) (4x y)3
(B) (u 3v)2 (2u v)(2u v)
• Application
EXAMPLE 8
Volume of a Cylindrical Shell
*A plastic water pipe with a hollow center is 100 inches long, 1 inch thick, and has
an inner radius of x inches (see the figure on the next page). Write an algebraic expression in terms of x that represents the volume of the plastic used to construct the pipe.
Simplify the expression. [Recall: The volume V of a right circular cylinder of radius
r and height h is given by V r 2h.]
*The symbol
denotes problems that are related to calculus.
1-2
Solution
1 inch
x inch
Polynomials: Basic Operations
21
A right circular cylinder with a hollow center is called a cylindrical shell. The volume of the shell is equal to the volume of the cylinder minus the volume of the hole.
Since the radius of the hole is x inches and the pipe is 1 inch thick, the radius of the
cylinder is x 1 inches. Thus, we have
cylinder Volume of
shell
Volume of
Volume of
hole
Volume (x 1)2 100 x 2100
100(x2 2x 1) 100x2
100x2 200x 100 100x2
100 inches
200x 100
Matched Problem 8
A plastic water pipe is 200 inches long, 2 inches thick, and has an outer radius of x
inches. Write an algebraic expression in terms of x that represents the volume of the
plastic used to construct the pipe. Simplify the expression.
Answers to Matched Problems
1. (A) 3x2 2x 1, x2 2xy y2
(B) 5, 3, 5
(C) 6, 4, 6
2. (A) 4u2 v2
(B) m3 3m2 2m 4
(C) x3 3x 2
4
3
2
2
3. 3x x 5x 2x 2
4. 3x 5x 10
5. 4x3 13x 6
6. (A) 8u2 2uv 3v2
(B) 4x2y2 9
(C) m2 16n2
(D) 4u2 12uv 9v2
(E) 36x2 12xy y2
7. (A) 2t2 4t 7
(B) 3u2 6uv 10v2
(C) 64x3 48x2y 12xy2 y3
8. Volume 200x2 200(x 2)2 800x 800
EXERCISE
1-2
A
10. 2(u 1) (3u 2) 2(2u 3)
11. 2y 3y[4 2( y 1)]
Problems 1–8 refer to the following polynomials:
(a) 2x3 3x2 x 5
(b) 2x2 x 1
(c) 3x 2
12. 4a 2a[5 3(a 2)]
1. What is the degree of (a)?
2. What is the degree of (b)?
13. (m n)(m n)
3. Add (a) and (b).
4. Add (b) and (c).
14. (a b)(a b)
5. Subtract (b) from (a).
6. Subtract (c) from (b).
15. (4t 3)(t 2)
7. Multiply (a) and (c).
8. Multiply (b) and (c).
16. (3x 5)(2x 1)
17. (3x 2y)(x 3y)
In Problems 9–28, perform the indicated operations and
simplify.
18. (2x 3y)(x 2y)
9. 2(x 1) 3(2x 3) (4x 5)
19. (2m 7)(2m 7)
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1 Basic Algebraic Operations
20. (3y 2)(3y 2)
21. (6x 4y)(5x 3y)
22. (3m 7n)(2m 5n)
23. (3x 2y)(3x 2y)
24. (4m 3n)(4m 3n)
25. (4x y)2
In Problems 53–56, perform the indicated operations and
simplify.
26. (3u 4v)2
27. (a b)(a2 ab b2)
53. 2(x 2)3 (x 2)2 3(x 2) 4
C
28. (a b)(a2 ab b2)
54. (2x 1)3 2(2x 1)2 3(2x 1) 7
55. 3x{x[x x(2 x)] (x 2)(x2 3)}
56. 2{(x 3)(x2 2x 1) x[3 x(x 2)]}
B
57. Show by example that, in general, (a b)2 a2 b2. Discuss possible conditions on a and b that would make this a
valid equation.
In Problems 29–42, perform the indicated operations and
simplify.
58. Show by example that, in general, (a b)2 a2 b2. Discuss possible conditions on a and b that would make this a
valid equation.
29. 2x 3{x 2[x (x 5)] 1}
30. m {m [m (m 1)]}
31. 2{3[a 4(1 a)] (5 a)}
59. If you are given two polynomials, one of degree m and the
other of degree n, m n, what is the degree of the sum?
32. 5b 3{ [2 4(2b 1)] 2(2 3b)}
33. (2x2 3x 1)(x2 x 2)
60. What is the degree of the product of the two polynomials in
Problem 59?
34. (x 3xy y )(x 3xy y )
2
2
2
2
61. How does the answer to Problem 59 change if the two polynomials can have the same degree?
35. (x 2y)2(x 2y)2
36. (n2 4nm m2)(n2 4nm m2)
62. How does the answer to Problem 60 change if the two polynomials can have the same degree?
37. (3u 2v)2 (2u 3v)(2u 3v)
38. (2a b)2 (a 2b)2
39. (z 2)(z2 2z 3) z 7
APPLICATIONS
40. (y 3)(y2 3y 1) 8y 1
41. (2m n)3
63. Geometry. The width of a rectangle is 5 centimeters less
than its length. If x represents the length, write an algebraic
expression in terms of x that represents the perimeter of the
rectangle. Simplify the expression.
42. (3a 2b)3
Problems 43–50 are calculus-related. Perform the indicated
operations and simplify.
64. Geometry. The length of a rectangle is 8 meters more than
its width. If x represents the width of the rectangle, write an
algebraic expression in terms of x that represents its area.
Change the expression to a form without parentheses.
43. 3(x h) 7 (3x 7)
44. (x h)2 x2
45. 2(x h)2 3(x h) (2x2 3x)
★
65. Coin Problem. A parking meter contains nickels, dimes,
and quarters. There are 5 fewer dimes than nickels, and 2
more quarters than dimes. If x represents the number of
nickels, write an algebraic expression in terms of x that represents the value of all the coins in the meter in cents. Simplify the expression.
★
51. Subtract the sum of the first two polynomials from the sum
of the last two: 3m2 2m 5, 4m2 m, 3m2 3m 2,
m3 m2 2
66. Coin Problem. A vending machine contains dimes and
quarters only. There are 4 more dimes than quarters. If x represents the number of quarters, write an algebraic expression in terms of x that represents the value of all the coins in
the vending machine in cents. Simplify the expression.
52. Subtract the sum of the last two polynomials from the sum
of the first two: 2x2 4xy y2, 3xy y2, x2 2xy y2,
x2 3xy 2y2
67. Packaging. A spherical plastic container for designer wristwatches has an inner radius of x centimeters (see the figure
on the next page). If the plastic shell is 0.3 centimeters
46. 4(x h)2 6(x h) (4x2 6x)
47. 2(x h)2 4(x h) 9 (2x2 4x 9)
48. 3(x h)2 5(x h) 7 (3x2 5x 7)
49. (x h)3 2(x h)2 (x3 2x2)
50. (x h)3 3(x h) (x3 3x)
1-3
Polynomials: Factoring
thick, write an algebraic expression in terms of x that represents the volume of the plastic used to construct the container. Simplify the expression. [Recall: The volume
V of a sphere of radius r is given by V 43r3.]
0.3 cm
68. Packaging. A cubical container for shipping computer
components is formed by coating a metal mold with polystyrene. If the metal mold is a cube with sides x centimeters
long and the polystyrene coating is 2 centimeters thick,
write an algebraic expression in terms of x that represents
the volume of the polystyrene used to construct the container. Simplify the expression. [Recall: The volume V of a
cube with sides of length t is given by V t3.]
SECTION
1-3
Does It Mean?
x cm
Figure for 67
Polynomials: Factoring
•
•
•
•
—What
• Factoring—
23
Factoring—What Does It Mean?
Common Factors and Factoring by Grouping
Factoring Second-Degree Polynomials
More Factoring
A factor of a number is one of two or more numbers whose product is the given
number. Similarly, a factor of an algebraic expression is one of two or more algebraic expressions whose product is the given algebraic expression. For example,
30 2 3 5
2, 3, and 5 are each factors of 30.
x 4 (x 2)(x 2)
(x 2) and (x 2) are each factors of x2 4.
2
The process of writing a number or algebraic expression as the product of other numbers or algebraic expressions is called factoring. We start our discussion of factoring
with the positive integers.
An integer such as 30 can be represented in a factored form in many ways. The
products
65
(12 )(10)(6)
15 2
235
all yield 30. A particularly useful way of factoring positive integers greater than 1 is
in terms of prime numbers.
DEFINITION 1
Prime and Composite Numbers
An integer greater than 1 is prime if its only positive integer factors are itself
and 1. An integer greater than 1 that is not prime is called a composite number. The integer 1 is neither prime nor composite.
Examples of prime numbers:
2, 3, 5, 7, 11, 13
Examples of composite numbers:
4, 6, 8, 9, 10, 12